Control systems have been widely used in a variety of applications, such as, for example, industrial manufacturing applications, navigational systems, communication systems, and information storage and retrieval systems. The control systems usually maintain a plant in a selected state (i.e., position, velocity, acceleration, temperature, pressure, or pH) despite unknown external inputs which may tend to force the plant to drift away from the selected state. The control systems may also cause the plant to change from one state to another state.
One method to change a plant from one state to another state is via the use of external controls. For example, maximum amplitude control signals (i.e., maximum current, voltage, temperature, torque, pressure, or flow) can be applied to the plant over time periods of specific lengths. For a system modeled as an Nth order linear system, the use of the maximum control signals over N consecutive, possibly equal, time periods can cause the plant to move from one arbitrary state to any other reachable state in the minimum possible time. The duration of these N consecutive time periods is a set of N real values, continuously distributed from zero to infinity. The specific length of these time periods is a continuous function of the parameters defining the plant. In order to further reduce the transition time of the plant from one state to another, the magnitude of the control signals can be increased.
In a digital control system, data is generally collected and processed at discrete time intervals. These time intervals relate to the times at which the digital control system collects data from a sensor measuring the state of the plant. The time intervals are typically called the sampling times of the control system. In response to these sampled measurements of the plant and to requests issued to the control system from a supervisory system, the control system generates updated commands (i.e., voltage, current, torque, pressure, flow, and temperature commands). These updated commands usually drive an actuator and cause a change in the state of the plant. These commands are typically generated at discrete time intervals. The rate at which the commands are generated and directed to the plant is often referred to as the update rate. In many cases, the update and sampling rates are equal. As a result, the digital controller can typically only take actions at discrete points in time.
The difficulty in utilizing a digital control system to achieve a truly minimal-time state transition is that the required switching times of the command signals are continuously distributed in time, whereas the digital control system typically take actions at discrete points in time. In addition, the digital control system may not have an update rate which can correlate with the required switching times for the possible or required state transitions. Furthermore, the time difference between the available update times are often very large relative to the switching precision required by the dynamics of the plant. Consequently, the commands of the digital control system have to be activated for longer or shorter periods of time than desired, resulting in an error in the final state (i.e., overshoot or undershoot of one or more states). In order to compensate for discrete update times, and still arrive at the desired final state, the control system must vary the magnitudes of control signals. Since the signals can not be increased, they must be lowered, and as a consequence, the signals must be applied for a longer time duration. As a result, the system is no longer truly time optimal since maximum command values are not used.